To me, writing it that way is perhaps a little awkward, and confusing. The writer is trying to represent a verbal pause to show that 'quote' is an aside, rather than part of the sentence. I would write it as: It'll be kept in the city's tourist areas to (quote) "show how classy Dubai is."

This sentence is not written properly, because there is no need to say "quote" when writing a sentence. Sometimes we say "quote" when talking, or "I am quoting", because the quotes " " are not visible when you are talking. But in writing you can see them, so there is no need to write "quote". They put a comma to allow a pause in the reading. Perhaps a dash would have been better, but as I said, in writing there is no need to say "quote". They are saying, in effect: It'll be kept in the city's tourist areas to "show how classy Dubay is". Mentioning that it is a quote causes the person reading or listening to understand that this is a quotation from something that somebody said beforehand, or it is a common saying.

The commas on both sides of "quote" are there to set off a non essential element of the sentence. "Quote" is unnecessary, and acts almost as an interjection, which also would be set off by commas. It is used in spoken English, not written, when the speaker wants the hearer to know that he is quoting something or someone.